共查询到20条相似文献,搜索用时 31 毫秒
1.
John Loverich Ammar Hakim & Uri Shumlak 《Communications In Computational Physics》2011,9(2):240-268
A discontinuous Galerkin method for the ideal 5 moment two-fluid plasma
system is presented. The method uses a second or third order discontinuous Galerkin
spatial discretization and a third order TVD Runge-Kutta time stepping scheme. The
method is benchmarked against an analytic solution of a dispersive electron acoustic
square pulse as well as the two-fluid electromagnetic shock [1] and existing numerical
solutions to the GEM challenge magnetic reconnection problem [2]. The algorithm can
be generalized to arbitrary geometries and three dimensions. An approach to maintaining
small gauge errors based on error propagation is suggested. 相似文献
2.
Simulations of Compressible Two-Medium Flow by Runge-Kutta Discontinuous Galerkin Methods with the Ghost Fluid Method
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Jianxian Qiu Tiegang Liu & Boo Cheong Khoo 《Communications In Computational Physics》2008,3(2):479-504
The original ghost fluid method (GFM) developed in [13] and the modified GFM (MGFM) in [26] have provided a simple and yet flexible way to treat two-medium flow problems. The original GFM and MGFM make the material interface
”invisible” during computations and the calculations are carried out as for a single
medium such that its extension to multi-dimensions becomes fairly straightforward.
The Runge-Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservation laws is a high order accurate finite element method employing the useful
features from high resolution finite volume schemes, such as the exact or approximate
Riemann solvers, TVD Runge-Kutta time discretizations, and limiters. In this paper,
we investigate using RKDG finite element methods for two-medium flow simulations
in one and two dimensions in which the moving material interfaces are treated via nonconservative methods based on the original GFM and MGFM. Numerical results for
both gas-gas and gas-water flows are provided to show the characteristic behaviors of
these combinations. 相似文献
3.
A RKDG Method for 2D Lagrangian Ideal Magnetohydrodynamics Equations with Exactly Divergence-Free Magnetic Field
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Shijun Zou Xiaolong Zhao Xijun Yu & Zihuan Dai 《Communications In Computational Physics》2022,32(2):547-582
In this paper, we present a Runge-Kutta Discontinuous Galerkin (RKDG)
method for solving the two-dimensional ideal compressible magnetohydrodynamics
(MHD) equations under the Lagrangian framework. The fluid part of the ideal MHD
equations along with $z$-component of the magnetic induction equation are discretized
using a DG method based on linear Taylor expansions. By using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of
the ideal MHD, an exactly divergence-free numerical magnetic field can be obtained.
The nodal velocities and the corresponding numerical fluxes are explicitly calculated
by solving multidirectional approximate Riemann problems. Two kinds of limiter are
proposed to inhibit the non-physical oscillation around the shock wave, and the second limiter can eliminate the phenomenon of mesh tangling in the simulations of the
rotor problems. This Lagrangian RKDG method conserves mass, momentum, and
total energy. Several numerical tests are presented to demonstrate the accuracy and
robustness of the proposed scheme. 相似文献
4.
A New Approach of High Order Well-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms
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Hyperbolic balance laws have steady state solutions in which the flux gradients are
nonzero but are exactly balanced by the source terms. In our earlier work [31–33], we designed
high order well-balanced schemes to a class of hyperbolic systems with separable source terms.
In this paper, we present a different approach to the same purpose: designing high order
well-balanced finite volume weighted essentially non-oscillatory (WENO) schemes and RungeKutta
discontinuous Galerkin (RKDG) finite element methods. We make the observation that
the traditional RKDG methods are capable of maintaining certain steady states exactly, if a
small modification on either the initial condition or the flux is provided. The computational
cost to obtain such a well balanced RKDG method is basically the same as the traditional
RKDG method. The same idea can be applied to the finite volume WENO schemes. We
will first describe the algorithms and prove the well balanced property for the shallow water
equations, and then show that the result can be generalized to a class of other balance laws.
We perform extensive one and two dimensional simulations to verify the properties of these
schemes such as the exact preservation of the balance laws for certain steady state solutions,
the non-oscillatory property for general solutions with discontinuities, and the genuine high
order accuracy in smooth regions. 相似文献
5.
A fully discrete discontinuous Galerkin method is introduced for solving
time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in
our scheme, discontinuous Galerkin methods are used to discretize not only the spatial
domain but also the temporal domain. The proposed numerical scheme is proved to be
unconditionally stable, and a convergent rate $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ is established under the $L^2$ -norm when polynomials of degree at most $r$ and $k$ are used for temporal and
spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ in
time step is observed numerically for the numerical fluxes w.r.t. temporal variable at
the grid points. 相似文献
6.
Three Discontinuous Galerkin Methods for One- and Two-Dimensional Nonlinear Dirac Equations with a Scalar Self-Interaction
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Shu-Cun Li & Huazhong Tang 《Communications In Computational Physics》2021,30(4):1150-1184
This paper develops three high-order accurate discontinuous Galerkin (DG)
methods for the one-dimensional (1D) and two-dimensional (2D) nonlinear Dirac
(NLD) equations with a general scalar self-interaction. They are the Runge-Kutta DG
(RKDG) method and the DG methods with the one-stage fourth-order Lax-Wendroff
type time discretization (LWDG) and the two-stage fourth-order accurate time discretization (TSDG). The RKDG method uses the spatial DG approximation to discretize
the NLD equations and then utilize the explicit multistage high-order Runge-Kutta
time discretization for the first-order time derivatives, while the LWDG and TSDG
methods, on the contrary, first give the one-stage fourth-order Lax-Wendroff type and
the two-stage fourth-order time discretizations of the NLD equations, respectively, and
then discretize the first- and higher-order spatial derivatives by using the spatial DG
approximation. The $L^2$ stability of the 2D semi-discrete DG approximation is proved
in the RKDG methods for a general triangulation, and the computational complexities of three 1D DG methods are estimated. Numerical experiments are conducted to
validate the accuracy and the conservation properties of the proposed methods. The
interactions of the solitary waves, the standing and travelling waves are investigated
numerically and the 2D breathing pattern is observed. 相似文献
7.
Lin Mu Junping Wang Xiu Ye & Shan Zhao 《Communications In Computational Physics》2014,15(5):1461-1479
A weak Galerkin (WG) method is introduced and numerically tested for the
Helmholtz equation. This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property. At the same time, the WG finite
element formulation is symmetric and parameter free. Several test scenarios are designed for a numerical investigation on the accuracy, convergence, and robustness of
the WG method in both inhomogeneous and homogeneous media over convex and
non-convex domains. Challenging problems with high wave numbers are also examined. Our numerical experiments indicate that the weak Galerkin is a finite element
technique that is easy to implement, and provides very accurate and robust numerical
solutions for the Helmholtz problem with high wave numbers. 相似文献
8.
9.
A Weighted Runge-Kutta Discontinuous Galerkin Method for 3D Acoustic and Elastic Wave-Field Modeling
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Numerically solving 3D seismic wave equations is a key requirement for
forward modeling and inversion. Here, we propose a weighted Runge-Kutta discontinuous Galerkin (WRKDG) method for 3D acoustic and elastic wave-field modeling. For this method, the second-order seismic wave equations in 3D heterogeneous anisotropic media are transformed into a first-order hyperbolic system, and
then we use a discontinuous Galerkin (DG) solver based on numerical-flux formulations for spatial discretization. The time discretization is based on an implicit diagonal Runge-Kutta (RK) method and an explicit iterative technique, which avoids
solving a large-scale system of linear equations. In the iterative process, we introduce
a weighting factor. We investigate the numerical stability criteria of the 3D method in
detail for linear and quadratic spatial basis functions. We also present a 3D analysis of
numerical dispersion for the full discrete approximation of acoustic equation, which
demonstrates that the WRKDG method can efficiently suppress numerical dispersion
on coarse grids. Numerical results for several different 3D models including homogeneous and heterogeneous media with isotropic and anisotropic cases show that the 3D
WRKDG method can effectively suppress numerical dispersion and provide accurate
wave-field information on coarse mesh. 相似文献
10.
An Adaptive Modal Discontinuous Galerkin Finite Element Parallel Method Using Unsplit Multi-Axial Perfectly Matched Layer for Seismic Wave Modeling
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Yang Xu Xiaofei Chen Wei Zhang & Xiao Pan 《Communications In Computational Physics》2022,31(4):1083-1113
The discontinuous Galerkin finite element method (DG-FEM) is a high-precision numerical simulation method widely used in various disciplines. In this paper, we derive the auxiliary ordinary differential equation complex frequency-shifted
multi-axial perfectly matched layer (AODE CFS-MPML) in an unsplit format and combine it with any high-order adaptive DG-FEM based on an unstructured mesh to simulate seismic wave propagation. To improve the computational efficiency, we implement
Message Passing Interface (MPI) parallelization for the simulation. Comparisons of
the numerical simulation results with the analytical solutions verify the accuracy and
effectiveness of our method. The results of numerical experiments also confirm the
stability and effectiveness of the AODE CFS-MPML. 相似文献
11.
Performance Analysis of a High-Order Discontinuous Galerkin Method Application to the Reverse Time Migration
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Caroline Baldassari Hé lè ne Barucq Henri Calandra Bertrand Denel & Julien Diaz 《Communications In Computational Physics》2012,11(2):660-673
This work pertains to numerical aspects of a finite element method based
discontinuous functions. Our study focuses on the Interior Penalty Discontinuous
Galerkin method (IPDGM) because of its high-level of flexibility for solving the full
wave equation in heterogeneous media. We assess the performance of IPDGM through
a comparison study with a spectral element method (SEM). We show that IPDGM is
as accurate as SEM. In addition, we illustrate the efficiency of IPDGM when employed
in a seismic imaging process by considering two-dimensional problems involving the
Reverse Time Migration. 相似文献
12.
A Simple Solver for the Two-Fluid Plasma Model Based on PseudoSpectral Time-Domain Algorithm
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Benoit Morel Remo Giust Kazem Ardaneh & Francois Courvoisier 《Communications In Computational Physics》2021,29(3):955-978
We present a solver of 3D two-fluid plasma model for the simulation of
short-pulse laser interactions with plasma. This solver resolves the equations of the
two-fluid plasma model with ideal gas closure. We also include the Bhatnagar-Gross-Krook collision model. Our solver is based on PseudoSpectral Time-Domain (PSTD)
method to solve Maxwell's curl equations. We use a Strang splitting to integrate Euler
equations with source term: while Euler equations are solved with a composite scheme
mixing Lax-Friedrichs and Lax-Wendroff schemes, the source term is integrated with
a fourth-order Runge-Kutta scheme. This two-fluid plasma model solver is simple to
implement because it only relies on finite difference schemes and Fast Fourier Transforms. It does not require spatially staggered grids. The solver was tested against
several well-known problems of plasma physics. Numerical simulations gave results
in excellent agreement with analytical solutions or with previous results from the literature. 相似文献
13.
Alan R. Schiemenz Marc A. Hesse & Jan S. Hesthaven 《Communications In Computational Physics》2011,10(2):433-452
A high-order discretization consisting of a tensor product of the Fourier collocation
and discontinuous Galerkin methods is presented for numerical modeling of
magma dynamics. The physical model is an advection-reaction type system consisting
of two hyperbolic equations and one elliptic equation. The high-order solution
basis allows for accurate and efficient representation of compaction-dissolution waves
that are predicted from linear theory. The discontinuous Galerkin method provides
a robust and efficient solution to the eigenvalue problem formed by linear stability
analysis of the physical system. New insights into the processes of melt generation
and segregation, such as melt channel bifurcation, are revealed from two-dimensional
time-dependent simulations. 相似文献
14.
A Runge Kutta Discontinuous Galerkin Method for Lagrangian Compressible Euler Equations in Two-Dimensions
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Zhenzhen Li Xijun Yu Jiang Zhu & Zupeng Jia 《Communications In Computational Physics》2014,15(4):1184-1206
This paper presents a new Lagrangian type scheme for solving the Euler
equations of compressible gas dynamics. In this new scheme the system of equations
is discretized by Runge-Kutta Discontinuous Galerkin (RKDG) method, and the mesh
moves with the fluid flow. The scheme is conservative for the mass, momentum and
total energy and maintains second-order accuracy. The scheme avoids solving the geometrical
part and has free parameters. Results of some numerical tests are presented
to demonstrate the accuracy and the non-oscillatory property of the scheme. 相似文献
15.
Weighted Interior Penalty Method with Semi-Implicit Integration Factor Method for Non-Equilibrium Radiation Diffusion Equation
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Rongpei Zhang Xijun Yu Jiang Zhu Abimael F. D. Loula & Xia Cui 《Communications In Computational Physics》2013,14(5):1287-1303
Weighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe time step limits, but also take advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation. 相似文献
16.
A High-Order Cell-Centered Discontinuous Galerkin Multi-Material Arbitrary Lagrangian-Eulerian Method
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Fang Qing Xijun Yu Zupeng Jia Meilan Qiu & Xiaolong Zhao 《Communications In Computational Physics》2020,28(4):1464-1501
In this paper, a high-order cell-centered discontinuous Galerkin (DG) multi-material arbitrary Lagrangian-Eulerian (MMALE) method is developed for compressible fluid dynamics. The MMALE method utilizes moment-of-fluid (MOF) interface
reconstruction technology to simulate multi-materials of immiscible fluids. It is an
explicit time-marching Lagrangian plus remap type. In the Lagrangian phase, an updated high-order discontinuous Galerkin Lagrangian method is applied for the discretization of hydrodynamic equations, and Tipton's pressure relaxation closure model
is used in the mixed cells. A robust moment-of-fluid interface reconstruction algorithm
is used to provide the information of the material interfaces for remapping. In the rezoning phase, Knupp's algorithm is used for mesh smoothing. For the remapping
phase, a high-order accurate remapping method of the cell-intersection-based type is
proposed. It can be divided into four stages: polynomial reconstruction, polygon intersection, integration, and detection of problematic cells and limiting. Polygon intersection is based on the "clipping and projecting" algorithm, and detection of problematic
cells depends on a troubled cell marker, and a posteriori multi-dimensional optimal order detection (MOOD) limiting strategy is used for limiting. Numerical tests are given
to demonstrate the robustness and accuracy of our method. 相似文献
17.
Runge-Kutta Discontinuous Galerkin Method Using WENO-Type Limiters: Three-Dimensional Unstructured Meshes
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This paper further considers weighted essentially non-oscillatory (WENO) and Hermite weighted essentially non-oscillatory (HWENO) finite volume methods as limiters for Runge-Kutta discontinuous Galerkin (RKDG) methods to solve problems involving nonlinear hyperbolic conservation laws. The application discussed here is the solution of 3-D problems on unstructured meshes. Our numerical tests again demonstrate this is a robust and high order limiting procedure, which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions. 相似文献
18.
Seshu Tirupathi Jan S. Hesthaven & Yan Liang 《Communications In Computational Physics》2015,18(1):230-246
Discontinuous Galerkin (DG) and matrix-free finite element methods with
a novel projective pressure estimation are combined to enable the numerical modeling
of magma dynamics in 2D and 3D using the library deal.II. The physical model
is an advection-reaction type system consisting of two hyperbolic equations to evolve
porosity and soluble mineral abundance at local chemical equilibrium and one elliptic
equation to recover global pressure. A combination of a discontinuous Galerkin
method for the advection equations and a finite element method for the elliptic equation
provide a robust and efficient solution to the channel regime problems of the
physical system in 3D. A projective and adaptively applied pressure estimation is employed
to significantly reduce the computational wall time without impacting the overall
physical reliability in the modeling of important features of melt segregation, such
as melt channel bifurcation in 2D and 3D time dependent simulations. 相似文献
19.
An Interface-Capturing Method for Resolving Compressible Two-Fluid Flows with General Equation of State
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T. S. Lee J. G. Zheng & S. H. Winoto 《Communications In Computational Physics》2009,6(5):1137-1162
In this study, a stable and robust interface-capturing method is developed
to resolve inviscid, compressible two-fluid flows with general equation of state (EOS).
The governing equations consist of mass conservation equation for each fluid, momentum
and energy equations for mixture and an advection equation for volume fraction
of one fluid component. Assumption of pressure equilibrium across an interface is
used to close the model system. MUSCL-Hancock scheme is extended to construct
input states for Riemann problems, whose solutions are calculated using generalized
HLLC approximate Riemann solver. Adaptive mesh refinement (AMR) capability is
built into hydrodynamic code. The resulting method has some advantages. First, it is
very stable and robust, as the advection equation is handled properly. Second, general
equation of state can model more materials than simple EOSs such as ideal and
stiffened gas EOSs for example. In addition, AMR enables us to properly resolve flow
features at disparate scales. Finally, this method is quite simple, time-efficient and easy
to implement. 相似文献
20.
Parameter Identification in Uncertain Scalar Conservation Laws Discretized with the Discontinuous Stochastic Galerkin Scheme
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Louisa Schlachter & Claudia Totzeck 《Communications In Computational Physics》2020,28(4):1585-1608
We study an identification problem which estimates the parameters of the
underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin
method, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertain
initial conditions and that a data set of an observed solution is given. The uncertainty is assumed to be uniformly distributed on an unknown interval and we focus on
identifying the correct endpoints of this interval. The first-order optimality conditions
from the discontinuous stochastic Galerkin discretization are computed on the time-continuous level. Then, we solve the resulting semi-discrete forward and backward
schemes with the Runge-Kutta method. To illustrate the feasibility of the approach,
we apply the method to a stochastic advection and a stochastic equation of Burgers' type. The results show that the method is able to identify the distribution parameters
of the random variable in the uncertain differential equation even if discontinuities are
present. 相似文献